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Minimal Lagrangian connections and related structures

A projective structure on a smooth manifold consists of an equivalence class $$\mathfrak{p}$$ of torsion-free connections on its tangent bundle, where two such connections are called equivalent if they have the same geodesics up to parametrisation.

The representative connections of a projective structure $$\mathfrak{p}$$ on a smooth manifold $$M$$ are in one-to-one correspondence with the sections of an affine bundle $$A \to M$$, whose total space carries a split-signature metric $$h_{\mathfrak{p}}$$ as well as a symplectic form $$\Omega_{\mathfrak{p}}$$, both of which are defined in a canonical fashion from $$\mathfrak{p}$$. Consequently, all the submanifold notions of (pseudo-)Riemannian geometry and symplectic geometry can be applied to the representative connections of $$\mathfrak{p}$$. This point of view gives rise to the notion of a minimal Lagrangian connection.

The aim of this research project is to investigate various questions related to minimal Lagrangian connections, in particular, to provide a characterisation of projective manifolds that arise from a minimal Lagrangian connection.

## Team Members

JProf. Dr. Thomas Mettler